# Area bounded by Point $P$ in xy plane, If $\max\left\{\bf{PA+PB\;,PB+PC}\right\}\leq 2,$

A point $$P$$ moves in $$xy$$ plane such that $$\max\left\{\bf{PA+PB\;,PB+PC}\right\}\leq 2,$$ Then Area

of the Regine Bounded by Point $$\bf{P}$$ is, If Coordinate of $$A(0,0)\;\;,B(1,0)$$ and $$C(2,0)$$ Given

$$\bf{My\; Try::}$$ Let Coordinate of Point $$P(x,y)\;,$$ Then We get

$$PA= \sqrt{x^2+y^2}\;\;\;\; ,PB=\sqrt{(x-1)^2+y^2}\;\;\;, PC = \sqrt{(x-2)^2+y^2}$$

Now Using Cases::

$$\bullet\; \max\left\{\bf{PA+PB\;,PB+PC}\right\}\leq 2\Rightarrow PA+PB\leq 2\;,$$ If $$PA+PB>PB+PC\Rightarrow PA>PC$$

$$\bullet\; \max\left\{\bf{PA+PB\;,PB+PC}\right\}\leq 2\Rightarrow PB+PC\leq 2\;,$$ If $$PA+PB\leq PB+PC\Rightarrow PA\leq PC$$

Now How can I solve after that, Help me

Thanks

Each of $\bf{PA+PB}$$\leq2 and \bf{PB+PC}$$\leq2$ describes an ellipse with foci of $\bf{A}$ and $\bf{B}$ for the first one and $\bf{B}$ and $\bf{C}$ for the second one. They both have semi-major radius of $a=1$ (as $2a=PF_1+PF_2$) and $b=\frac{\sqrt3}{2}$ (as $f=\sqrt{a^2-b^2}$ and $f=\frac12$).